Why it's called the error function

Error integral

 

The Gaussian error integral (after Carl Friedrich Gauß) is the distribution function of the standard normal distribution. It is common with and is the integral of to via the density function of the normal distribution with and . Since the total area below the density curve (also called the Gaussian bell) is equal to 1, the value of the error integral is for also 1 (see section normalization).

definition

The error integral is through

Are defined.

If you leave the integral at first instead of begin, one speaks of :

Relation to the Gaussian error function

Through the substitution in the above formulas and by appropriate transformations can be omitted or. the error function

or.

derive.

application

The error integral indicates the probability that a standard normally distributed random variable has a value less than or equal to accepts. Conversely, the probability for a value can be greater or equal can be determined by forms.

As an electrotechnical example, let us assume a Gaussian-distributed interference noise from the scattering assumed that a transmission channel is superimposed. This channel works flawlessly as long as the interference is in the range of −5 V ... +5 V. The question of how likely a faulty transmission is:

Probability for a noise value not greater than -5 V:

Probability for a noise value at least equal to +5 V:

The overall probability of a transmission error then results from

Normalization

About the standardization to prove, we charge

Even if no antiderivative of the integrand can be expressed as an elementary function, there are still more than half a dozen possible solutions to determine its value, starting with De Moivre's first approximations from 1733 to the work of Laplace and Poisson from around 1800 to towards a completely new solution approach by SP Eveson from 2005. One of the decisive tricks for his calculation (allegedly by Poisson) is to switch to a higher dimension and to parameterize the resulting 2D integration area differently:

The basis for the first transformation is the linearity of the integral.

Instead of along Cartesian coordinates, now integrated along polar coordinates, what the substitution and it corresponds to, and one finally obtains with the transformation theorem

With this we get:

Based on an article in: Wikipedia.de
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Date of the last change: Jena, 20.06. 2020